Completeness in supergravity constructions

0.25 MB·English

by  V. Cortes

Completeness in supergravity constructions
V. Cort´es1, T. Mohaupt2 and H. Xu1
1Department of Mathematics
1
1 and Center for Mathematical Physics
0 University of Hamburg
2
Bundesstraße 55, D-20146 Hamburg, Germany

n
[email protected]
a
J
6 2Theoretical Physics Division
2
Department of Mathematical Sciences

] University of Liverpool
h
t Liverpool L69 3BX, UK
-
p [email protected]
e
h
January 26, 2011
[

1
v
3
0 Abstract
1
5
. We prove that the supergravity r- and c-maps preserve completeness. As a con-
1
0 sequence, any component H of a hypersurface h = 1 defined by a homogeneous
{ }
1 cubic polynomial h such that ∂2h is a complete Riemannian metric on H defines
1 −
a complete projective special Ka¨hler manifold and any complete projective special
:
v Ka¨hler manifold defines a complete quaternionic Ka¨hler manifold of negative scalar
i
X curvature. We classify all complete quaternionic Ka¨hler manifolds of dimension less
r or equal to 12 which are obtained in this way and describe some complete examples
a
in 16 dimensions.
Contents
1 Completeness of metrics on product manifolds and bundles 4
1.1 Generalisation to the case of bundles . . . . . . . . . . . . . . . . . . . . . 7
2 The generalized supergravity r-map preserves completeness 7
3 The supergravity c-map preserves completeness 10
1
4 Examples: Complete quaternionic K¨ahler manifolds associated to cubic
polynomials 13
4.1 Classification of complete cubic curves and corresponding 12-dimensional
quaternionic Ka¨hler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Examples of complete cubic surfaces and corresponding 16-dimensional
quaternionic Ka¨hler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Globalisation of the Ferrara-Sabharwal metric 17
5.1 From Griffiths to Weil flags in special Ka¨hler geometry . . . . . . . . . . . 20
Introduction
The supergravity r-map and the supergravity c-map are geometric constructions known
to theoretical physicists working in supergravity and string theory. They can be obtained
by dimensional reduction of the vector multiplet sector of supergravity theories with eight
real supercharges from 5 to 4 and from 4 to 3 spacetime dimensions, respectively. The
reduction from 4 to 3 dimensions was worked out by Ferrara and Sabharwal [FS], who
used the resulting explicit local description of the c-mapto prove that it maps a projective
special Ka¨hler manifold (see Definition 4) of real dimension 2n defined by a holomorphic
prepotential F = F(z0,...,zn) homogeneous of degree two to a quaternionic Ka¨hler
manifold of dimension 4n+4 of negative scalar curvature. Similarly, it was shown by de
Wit and Van Proeyen [DV2] that the r-map maps projective special real manifolds (see
Definition 1) of dimension n defined by a cubic polynomial h = h(x1,...,xn) to projective
special Ka¨hler manifolds of dimension 2n+2.
Despite some recent advances in the geometric understanding of the r-map [AC2] and
the c-map [H], as well as in finding new formulations of the c-map within the formalisms
of supergravity [DDKV, DRV, DS, RVV], very little is know about global geometric
properties of these constructions. In recent approaches to hypermultiplet moduli spaces,
there have been efforts aiming at the computation of quantum corrections to the Ferrara-
Sabharwal metric inspired by the work of Gaiotto, Moore and Neitzke on wall crossing
[GMN]. To our present knowledge this has been successfull only in the limit in which
gravity decouples, that is in the framework of hyper-K¨ahler geometry. The problem of
completeness of quaternionic Ka¨hler metrics obtained as deformations (e.g. by quantum
corrections) of quaternionic Ka¨hler metrics constructed by the c-map is an interesting
subject for future investigation.
2
The main results of this paper are Theorem 4 and Theorem 5, which state that the
supergravity r-map and the supergravity c-map preserve completeness. As a consequence,
we obtain an effective method for the construction of complete projective special Ka¨hler
and quaternionic Ka¨hler manifolds starting from certain real affine hypersurfaces defined
by homogeneous cubic polynomials h. Any such hypersurface of dimension n defines a
quaternionic Ka¨hler manifold of dimension 4n + 8, that is a point defines an 8-fold, a
curve defines a 12-fold, a surface defines a 16-fold etc. The study of the completeness
of the quaternionic Ka¨hler manifold is reduced to that of the completeness of the cubic
hypersurface H Rn+1 with respect to the Riemannian metric ∂2h H. We show how
⊂ − |
to obtain interesting complete examples and even classification results in low dimensions.
In particular, we find two complete inhomogeneous examples, see Corollary 4 b) and
Example 3. The homogeneous examples are automatically complete, as is any complete
Riemannian manifold. Moreover, all known examples of homogeneous projective special
Ka¨hler manifolds with exception of the complex hyperbolic spaces are in the image of
the r-map and all known examples of homogeneous quaternionic Ka¨hler manifolds of
negative scalar curvature with exception of the quaternionic hyperbolic spaces are in the
image of the c-map, see [DV2] and references therein. The known examples include the
homogeneous projective special Ka¨hler manifolds of semisimple groups classified in [AC1]
andthequaternionic Ka¨hler manifoldsadmitting a simply transitive (completely) solvable
groupofisometries classified in[A, C1]. The first class contains only Hermitiansymmetric
spaces of noncompact type, whereas the second class contains all quaternionic Ka¨hler
symmetric spaces of noncompact type as well as all known nonsymmetric homogeneous
quaternionic Ka¨hler manifolds. We plan to work towards general completeness results for
hypersurfaces in higher dimensions in the future.
One of the inhomogeneous complete examples is the ‘quantum STU model’ [AFT,
AFGNT, KV, KKLMV]. This model, or actually family of models, can be constructed by
compactification of the heterotic string on manifolds with holonomy contained in SU(2)
and with instanton numbers (12,12) or (13,11) or (10,4). The special real, special Ka¨hler
andspecial quaternionic manifoldoccur asmoduli spaces of compactifications onK3 S1,
×
K3 T2 and K3 T3, respectively. The qualification ‘quantum’ refers to the quantum
× ×
corrections to the cubic part of the underlying Hesse potential
1
h = STU h = STU + U3 ,
→ 3
which makes the corresponding manifolds inhomogeneous. This deformation is captured
by the triple intersection formsof dual models, which arecompactifications on Calabi-Yau
threefolds which areelliptic fibrationsover theHirzeburch surfaces F , F , F , respectively
0 1 2
[KV,MV,W]. Thechoiceofthebaseofthefibrationcorrespondstothechoiceofinstanton
3
numbers in the dual heterotic model. The heterotic compactifications on K3 T2 are
×
dual to type-IIA compactifications on the corresponding Calabi-Yau threefolds, and the
vector multiplet moduli spaces of these models are the complexified Ka¨hler cones of the
Calabi-Yau threefolds. The type-IIA model has an ‘M-theory lift’ to a compactification
of eleven-dimensional M-theory on the same Calabi-Yau threefold, which is then dual
to the heterotic compactification on K3 S1. The moduli spaces of five-dimensional
×
vector multiplets correspond to a fixed volume slice of the (real) Ka¨hler cones of the
underlying Calabi-Yau threefolds. Some properties of the special Ka¨hler and special real
metrics occuring in the vector multiplet sectors of these models have been discussed in
the physics literature [LSTY, ChEA, MM]. The geodesically complete spaces considered
in this paper are the natural choices of scalar manifolds if these models are considered
as supergravity models. The moduli spaces relevant for string theory are sub-domains of
these manifolds, as discussed in [MV, W, LSTY, ChEA, MM].
In the last section we give a geometric interpretation of the complex (n+1) (n+1)-
×
matrix N = (N ), which defines the nontrivial part of the Ferrara-Sabharwal metric. We
IJ
show that N defines a Weil flag which is precisely the image of the Griffiths flag associated
with the variation of Hodge structure of weight 3 defined by the underlying affine special
Ka¨hler manifold under a natural Sp(R2n+2)-equivariant map from Griffiths to Weil flags,
see Corollary 5. Furthermore, N is canonically associated with a positive definite metric
which differs from the (indefinite) affine special Ka¨hler metric by a canonical sign switch,
see Corollary 6. Using this geometric insight, we are able to extend the c-map and our
completeness result to special Ka¨hler manifolds which do not admit a global description
by a single prepotential, see Theorem 10.
1 Completeness of metrics on product manifolds and
bundles
Let us recall that a Riemannian manifold (M,g) is called complete if it is complete as a
metricspace, i.e.ifeveryCauchysequenceinM converges. ThebasicresultinRiemannian
geometry concerning completeness is the following theorem of Hopf and Rinow, cf. [O]
Ch. 5, Thm. 21.
Theorem 1 (Hopf-Rinow) For a Riemannian manifold (M,g) the following condi-
tions are equivalent:
(i) M is complete.
(ii) M is geodesically complete, i.e. every inextendible geodesic ray has infinite length.
4
(iii) Any closed and bounded subset of M is compact.
We give another equivalent formulation of completeness in terms of (smooth) curves
γ : I M, where I R is a (nondegenerate) interval.
→ ⊂
Lemma 1 A Riemannian manifold (M,g) is complete if and only if every curve γ :
I M which is not contained in any compact subset of M has infinite length.
→
Proof: “ ” Let us assume that M is complete and that γ : I M is not contained in
⇒ →
any compact set. Then γ(I) M is unbounded, since it is not contained in any ball.
⊂
Here we use the fact that closed balls are compact in any complete Riemannian manifold,
by Theorem 1 (iii). Clearly, an unbounded curve has infinite length.
“ ” If M is not complete, then there exists an inextendible geodesic ray γ : [0,L) M
⇐ →
of finite length L, by Theorem 1 (ii). The ray γ is not contained in any compact set K,
because otherwise γ(t) would converge to a limit point in K for t L and γ could then
→
˜ ˜
be extended to a geodesic ray γ˜ : [0,L) M for some L > L.
→
Let M = M M be a product manifold and denote by π : M M , i = 1,2, the
1 2 i i
× →
projections. We consider Riemannian metrics g of the form
g = g +g , (1.1)
1 2
where g is (the pullback of) a Riemannian metric on M and g Γ(π S2T M ) is a
1 1 2 ∈ 2∗ ∗ 2
family of Riemannian metrics on M depending on a parameter p M . Notice that the
2 1
∈
tensors g and g take the form
1 2
g = g(1)(x)dxadxb, g = g(2)(x,y)dyαdyβ,
1 ab 2 αβ
X X
with respect to local coordinates x = (xa) on M and y = (yα) on M .
1 2
We will assume that the tensor field g satisfies the following condition:
2
(C) For all compact subsets A M there exists a complete Riemannian metric g on
1 A
⊂
M such that g π g on A M M.
2 2 ≥ 2∗ A × 2 ⊂
Lemma 2 Assume that (M ,g ) is complete and that g satisfies the condition (C).
1 1 2
Then (M,g) is complete.
Proof: Let γ = (γ ,γ ) : I M = M M be a curve which is not contained in any
1 2 1 2
→ ×
compact subset K M. In view of Lemma 1, it suffices to show that any such curve γ
⊂
has infinite length. From the assumption on γ it follows that
5
(i) γ : I M is not contained in any compact subset K M or
1 1 1 1
→ ⊂
(ii) γ : I M is not contained in any compact subset K M .
2 2 2 2
→ ⊂
(Otherwise, γ would be contained in a compact set K = K K .) In case (i), L(γ ) = ,
1 2 1
× ∞
by the completeness of (M ,g ) and Lemma 1. Comparing lengths we obtain
1 1
L(γ) L(γ ) =
1
≥ ∞
and, hence, L(γ) = . If (i) is not satisfied, then γ (I) is contained in a compact set
1
∞
A = K M . By the completeness of (M ,g ) and property (ii), we now have that
1 1 2 A
⊂
L (γ ) = . Now it suffices to compare the lengths: L(γ) L (γ ) = .
gA 2 ∞ ≥ gA 2 ∞
Theorem 2 Let (M ,g ) be a complete Riemannian manifold and (g (p)) a smooth
1 1 2 p
family of G-invariant Riemannian metrics on a homogeneous manifold M = G/K, de-
2
pendingona parameterp M . Thenthe Riemannianmetricg = g +g on M = M M
1 1 2 1 2
∈ ×
is complete. Moreover, the action of G on M induces an isometric action of G on (M,g).
2
Proof: The last assertion is obvious. For the completeness of (M,g), it suffices to check
that g satisfies the condition (C) of Lemma 2. We use the natural one-to-one correspon-
2
dence between G-invariant Riemannian metrics on M = G/K and K-invariant scalar
2
products on the vector space T M = g/k, o = eK. Under this correspondence, the family
o 2 ∼
(g (p)) corresponds to a family (β(p)) of scalar products. For every compact subset
2 p p
A M the family (β(p)) is uniformly bounded from below by a scalar product β :
1 p A A
⊂ ∈
β(p) β , for all p A.
A
≥ ∈
This implies
g (p) g , for all p A,
2 A
≥ ∈
for the G-invariant Riemannian metric g associated with β . Now it suffices to remark
A A
that g is complete, as is any G-invariant Riemannian metric on G/K.
A
Corollary 1 Let g = g dxadxb be a complete Riemannian metric on a domain
U ab
U Rn. Then the metric
⊂ P
3
g = g (x)(dxadxb +dyadyb) (1.2)
M ab
4
X
on M = U Rn is complete. The action of Rn by translations in the y-coordinates is
×
isometric and the projection M U is a principal fiber bundle with structure group Rn.
→
The submanifold U = U 0 U Rn = M is totally geodesic.
×{ } ⊂ ×
6
(Thefactor 3 isintroducedonlyinordertoobtaintheusualnormalizationoftheprojective
4
special Ka¨hler metric for the r-map defined in the next section.)
Proof: For the completeness it suffices to apply Theorem 2 to the case M = G = Rn,
2
g = g (x)dyadyb. U M is totally geodesic as the fixed point set of the isometric
2 ab
⊂
involution (x,y) (x, y).
P 7→ −
1.1 Generalisation to the case of bundles
More generally, for later applications we consider now a bundle π : M M with
1
→
standard fiber M over a Riemannian manifold (M ,g ). We suppose that the total
2 1 1
space M is endowed with a Riemannian metric g such that for all p M there exists a
1
∈
neighbourhood U M and a local trivialisation π 1(U) = U M with respect to which
1 − ∼ 2
⊂ ×
the metric takes the form
g|π−1(U) = g1|U +g2U, (1.3)
where gU is a smooth family of Riemannian metrics on M depending on a parameter
2 2
in U. Such metrics g will be called bundle metrics. We will assume that gU satisfies the
2
condition (C) for all compact subsets A U. Lemma 2 and Theorem 2 have the following
⊂
straightforward generalisations:
Lemma 3 Assume that (M ,g ) is complete and that the local fiber metrics gU in
1 1 2
equation (1.3) satisfy the condition (C) for all compact subsets A U. Then the bundle
⊂
metric g is complete.
Theorem 3 Let g be a bundle metric on a bundle π : M M over a complete
1
→
Riemannian manifold (M ,g ) and assume that the standard fiber is a homogeneous space
1 1
M = G/K and that the fiber metrics gU are G-invariant. Then (M,g) is complete.
2 2
2 The generalized supergravity r-map preserves com-
pleteness
Let U Rn be a domain which is invariant under multiplication by positive numbers and
⊂
let h : U R>0 be a smooth function which is homogeneous of degree d R 0,1 .
→ ∈ \ { }
Then
H := x U h(x) = 1 U
{ ∈ | } ⊂
is a smooth hypersurface and U = R>0 H = R>0 H. We will assume that 1∂2h is
· ∼ × −d
positive definite on TH. This easily implies that 1∂2h is a Lorentzian (if d > 1) or
−d
Riemannian (if d < 1) metric on U which restricts to a Riemannian metric gH on H.
7
Definition 1 The Riemannian manifold (H,gH) is called a projective special real man-
ifold if, in addition, h is a polynomial function of degree d = 3.
Example Let V = H1,1(X,R) be the (1,1)-cohomology of a compact Ka¨hler manifold
X of complex dimension d 2 and U V the Ka¨hler cone. We define a homogeneous
≥ ⊂
polynomial h of degree d on V by
h(a) = a d = a a, a V.
∪
∪···∪ ∈
The polynomial h defines a positive function on U and the metric gH defined above is
positive definite on the hypersurface
H := x U h(x) = 1 V = Rn, n = h1,1(X).
∼
{ ∈ | } ⊂
This follows from the Hodge-Riemann bilinear relations, see [We] Chap. V, Sec. 6, which
imply that gH is positive definite on the primitive cohomology H01,1(X,R) = TκH defined
as the kernel of the cup product with κd 1 : H2(X,R) H2d(X,R), where κ U is a
−
→ ∈
Ka¨hler class. If d = 3 then (H, gH) is a projective special real manifold.
We endow U with the Riemannian metric
1
g = ∂2lnh (2.1)
U
−d
and M = TU = U Rn with the metric (1.2).
∼
×
Proposition 1 (U,gU) is isometric to the product (R H,dr2 +gH).
×
Proof: This is a straightforward calculation using the diffeomorphism
R H U, (r,x) erx,
× → 7→
and the formula (2.1).
Definition 2 The correspondence (H,gH) (M,gM) is called the generalized super-
7→
gravity r-map. The restriction to polynomial functions h of degree d = 3 is called the
supergravity r-map.
Next we need to recall the notion of a projective special Ka¨hler manifold [ACD, DV1, F].
It is best explained starting from the notion of a conical special Ka¨hler manifold, cf.
[ACD, CM]. For the purpose of this paper, we shall restrict the signature of the metric
by the condition (iv) in the following definition.
8
Definition 3 A conical special K¨ahler manifold (M,J,g, ,ξ) is a pseudo-K¨ahler man-
∇
ifold (M,J,g) endowed with a flat torsionfree connection and a vector field ξ such
∇
that
(i) ω = 0, where ω = g( ,J ) is the K¨ahler form,
∇ · ·
(ii) d J = 0, where J is considered as a one-form with values in the tangent bundle,
∇
(iii) ξ = Dξ = Id, where D is the Levi-Civita connection,
∇
(iv) g is positive definite on the plane D = span ξ,Jξ and negative definite on D .
⊥
{ }
It is shown in [ACD, CM] that the geometric data of a conical special Ka¨hler manifold
can be locally encoded in a holomorphic function F homogeneous of degree 2 defined in
a domain M Cn, see Theorem 2 and Proposition 6 [CM]. In fact, F is the generating
F
⊂
function of a holomorphic Lagrangian immersion M C2n, which induces on M the
F F
→
structure of a conical special Ka¨hler manifold. F is called the holomorphic prepotential
in the supergravity literature [DV1]. Under the assumptions of Definition 3, the vector
fields ξ and Jξ define a holomorphic action of a two-dimensional Abelian Lie algebra. We
will assume that this infinitesimal action lifts to a principal C -action on M with the base
∗
manifold M¯ = M/C . Then the hypersurface S := p M g(ξ(p),ξ(p)) = 1 M is
∗
{ ∈ | } ⊂
an S1-principal bundle over M¯. The principal action is isometric, since it is generated
by the Killing vector field Jξ. Therefore, the Lorentzian metric g = g induces a
S S
− |
Riemannian metric g¯ on M¯, which is easily seen to be Ka¨hlerian. In fact, the negative
definite Ka¨hler manifold (M¯, g¯) is precisely the Ka¨hler reduction of (M,g) with respect
−
to the above isometric Hamiltonian S1-action for a positive level of the moment map,
which is µ = g(ξ,ξ).
2
¯
Definition 4 The K¨ahler manifold (M,g¯) is called a projective special K¨ahler manifold.
The metric g¯ is called a projective special K¨ahler metric.
A standard example of a projective special Ka¨hler metric is the Weil-Petersson metric on
the space of complex structure deformations of a Calabi-Yau 3-fold, that is of a compact
Ka¨hler manifold with holonomy SU(3), see, for instance, [T, C2].
Theorem 4 The generalized supergravity r-map maps complete Riemannian manifolds
(H,gH) as above to complete K¨ahler manifolds (M,gM) with a free isometric action of the
vector group Rn. The supergravity r-map maps complete projective special real manifolds
to complete projective special K¨ahler manifolds.
9
Proof: The isometric action of Rn exists for general metrics as in Corollary 1. The Ka¨hler
property follows from the fact that the metrics g considered here are of Hessian type, cf.
U
[AC2] Prop. 3. Infact, (H,gH) is mappedto M = U Rn ∼= Rn+iU Cn with the metric
× ⊂
g defined by the Ka¨hler potential lnh(x), where (x,y) U Rn = M is identified
M
− ∈ ×
with ζ = y + ix Rn + iU Cn. If h is a cubic polynomial then a simple calculation
∈ ⊂
shows that the metric g can be also obtained from the Ka¨hler potential K(1,ζ) defined
M
by
n
K(z) = ln i zIF¯ F z¯I ,
I I
− −
!
I=0
X(cid:0) (cid:1)
where
F(z0,...,zn) = h(z1,...,zn)/z0 (2.2)
is a holomorphic function homogeneous of degree 2 on the domain
M˜ := z0 (1,ζ) z0 C , ζ Rn +iU Cn+1.
∗
{ · | ∈ ∈ } ⊂
(It suffices to check that i (zIF¯ F z¯I) = 8h(x).) This shows that (M,g ) is
z02 I − I M
| |
a projective special Ka¨hler manifold with the holomorphic prepotential F. The corre-
P
sponding conical special Ka¨hler manifold is the C -bundle M˜ M endowed with the
∗
→
affine special Ka¨hler metric g = 2 n (ImF )dzIdz¯J, which has signature (2,2n).
M˜ I,J=0 IJ
By Proposition 1 and Corollary 1, thePcompleteness of (H,gH) implies that of (U,gU) and
the completeness of (U,g ) implies that of (M,g ).
U M
Corollary 2 Any complete projective special real manifold (H,gH) of dimension n 1
− ≥
0 admits a canonical realisation as a totally geodesic submanifold of a complete projective
special K¨ahler manifold (M,g ) with a free isometric action of the group Rn. Each orbit
M
of Rn is flat and and intersects the submanifold H M orthogonally in exactly one point.
⊂
Proof: The orbits are flat since the metric is translation invariant. The hypersurface
H U = R>0 H is totally geodesic in virtue of Proposition 1 and U M is totally
∼
⊂ × ⊂
geodesic in virtue of Corollary 1. Therefore H M is totally geodesic.
⊂
Notice that the above proof shows that H M = r(H) is totally geodesic also for the
⊂
generalized r-map.
3 The supergravity c-map preserves completeness
A projective special Ka¨hler manifold (M,g ) which is globally defined by a single holo-
M
morphic prepotential F is called a projective special K¨ahler domain. Notice that the
10